Towards finishing off the axiom of reducibility

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Towards finishing off the axiom of reducibility

Philosophia Scientiæ, tome 1, n^o3 (1996), p. 17-35

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Philippe de Rouilhan

Institut d'Histoire et Philosophie des Sciences et des Techniques

Abstract. This article is about Russell's theory of types and, more precisely, about the axiom of reducibility. Since this axiom appeared, none of the criticisms it has been subjected to by Russell himself, then by Poincaré, Wittgenstein, Chwistek, etc., invalidâtes it except, it seems, that of Gôdel [1944]. But Gôdel's criticism is informai and dogmatic. I propose a formalization of this criticism and an argument from analogy in favor of it; along the way, I réfute the criticism Charles Parsons has endeavored to make of Gôdel criticism in volume II of Gôdel's Collected Works. This, of course, présupposes the prior formalisation of Russeirs theory of types itself.

Résumé. Cet article porte sur la théorie russellienne des types, et plus précisément sur l'axiome de réductibilité. Aucune des critiques dont il a été l'objet de la part de Russell lui-même, puis de Poincaré, de Wittgenstein, de Chwistek, etc., depuis son apparition n'est dirimante sauf, semble-t-il, celle de Gôdel [1944]. Mais la critique de Gôdel reste informelle et dogmatique. Je proposerai une formalisation de cette critique et un argument analogique en sa faveur ; et je réfuterai, au passage, la critique que, dans le volume II des Collected Works de Gôdel, Ch. Parsons croit pouvoir faire de la critique gôdélienne. Cela suppose, évidemment, une formalisation préalable de la théorie russellienne des types elle-même.

1. The Vicious Circle Principle, the Notion of Order

Most scholars hâveemphasized the 'appalling complexity', the 'overwhelming prolifération' involved in the Russellian hierarchy of types, its 'labyrinthine' character. It must hâve been of the 'beastly theory of types', as Wittgenstein [1913a] termed it, that Russell was at times thinking when he cried out in the course of their conversations: "Logic's hell!" [Wittgenstein 1977, 30]. However, when it cornes to the theory of types, it is Russell's approach, more than the thing itself, which is beastly. In his article on Russell's logic, Gôdel decried the lack of 'formai précision' of Principia Mathematica, and the absence, "above ail, [of] a précise statement of the syntax of formalisai" [Gôdel 1944, 126]. The work of logicians, that of Schutte [1960, §27] or of Church [1976], for example, has shown well that this kind of theory can be mastered on the formai level. The problem has remained, however, of mastering RusselVs theory, and not just the 'syntax of formalism', as Gôdel wished, but its intended interprétation as well.

Before being presented on May 17, 1994 at the Congrès International Henri Poincaré, Nancy, May 14-18, 1994, the content of this paper was fîrst presented at the May 3, 1994 meeting of the Séminaire d'épistémologie comparative, Aix- en-Provence, upon the invitation of Prof. E. Schwartz. A much more detailed présentation is to be found in my Russell et le cercle des paradoxes [Rouilhan 1996], especially chap. VI, §§ 11-14 and 18-23]. I thank C. Hill for her help in putting the présent version into English.

The theory I will présent is the resuit of a certain compromise made regarding some not very easily reconcilable demands: that of remaining historically faithful to Russell, but also that of theoretical consistency, and yet again that of placing Russell's theory with a historical perspective which makes it intelligible. This theory will be a theory of individuals, propositions, and (propositional) functions. It will satisfy the vicious circle principle in the technical form Russell gave it in 1908:

Whatever contains an apparent variable must not be a possible value of that variable. [Russell 1908, 237; 1956, 75]¹

But I will extend the idea of apparent variable further than Russell did, extending it to the variables which stand for possible arguments in a function and, in Russell's notation, are sometimes indicated by a circumflex. This will allow me to integrate into the vicious circle principle, as Russell understood it, the principle regarding functions he was led to add to it.

Putting the vicious circle principle into use involves the idea of order. Let us, as Russell sometimes did in heuristic présentations of the vicious circle principle, talk in terms of 'presupposition'. An entity 'présupposes' the values of its apparent variables. This is why it cannot be one of thèse values. It also 'présupposes' the values of the apparent variables thèse values may contain and cannot be one of thèse new values either. Etc. The idea is to assign an 'order' n e N to each entity, measuring, so to speak, its 'depth of presupposition', and to comply with the vicious circle principle by having apparent variables take values of orders strictly lower than those of the entities in which they occur. In particular, the arguments of a function will be of strictly lower orders than that of the function. Moreover, to remain faithful to Russell, not only will the orders of the arguments hâve an upper bound . For each place of the argument, the order will also be determinate. One is finally led to define the order of an entity as follows: the order of an individual will be 0; the order of a proposition or function will be the least whole number greater than or equal to the orders of its constants (in the extra-linguistic sensé), and strictly greater than the orders (of the values) of its variables. As concerns language, the order of a symbol for an entity, or entities (if this symbol contains real variables), will be that of the corresponding entities, and this will be the least whole number greater than or equal

The notion of apparent (bound) variable must be understood hère, with Russell, in an extra-ordinary, extra-linguistic sensé. An entity contains an apparent variable in this sensé if its logically perfecl expression contains one in the ordinary, linguistic sensé. The notion of real (free) variable is obviously not capable of playing this dual rôle.

to the orders of its constants and of its real variables, and strictly greater than those of its apparent variables.

2. The Notion of Type, the Hierarchy of Types

As for principles for the Russellian hierarchy of individuals, propositions and functions, there was not only the vicious circle principle, but also a principle analogous to the one which had led Frege to his own hierarchy of functions. The notion of order was only one of the notions involved in the notion of 'type'. Trying to remain faithful, in ail essentials, to Russell's thought, one is led to assign to individuals a certain type, say i; to propositions of the nth order a certain type determined by this order alone, say nn\ and to a function of the n * order a certain type determined by this order and the types rj, ..., rk of its arguments, say («, rl s..., rk ).

The set TT of the types in question may be defined as the smallest set X with a function of order od: X —» N such that (for ail n > 0, ail k > 1, and ail r j , . . . , rk)

(i) i, nn e X\ od(i) = 0, od(7Cn) = n;

(ii) (rj, ..., rk e X and n > 1 + m a x j ^ ^ od(r£)) =>

«n, r]9..., rk) G X and od««, rh ..., rk)) = n).

Each type r, therefore, has its order od(r), and one has:

od(i) = 0, od(7C„) = n,

od((n, r1? ..., rk)) = n > 1 + max1SKfc od(rj)).

The types can be compared in terms of their orders, and a structure of complète quasi-order given to the set Tr in this way. This is what is called the hierarchy of types.

Naturally, the so-called theory of types is not the theory having as its objects the types just discussed, but the theory of individuals, propositions and functions set out in accordance with the hierarchy of types just discussed.

3. Towards the Formalization of the Theory of Types: the Language L(TTR)

L(TTR) will be a language with infinitely many sorts of

variables, namely, for every r e Tr, variables ' xx ', ' x2 \ * x3 \ ... Gf entities of type r: variables of individuals if r = i, variables of propositions if r is of the form pn, and variables of functions if r is of the form («, rL, ..., rk ). Besides the variables may occur constants, namely, for any r G Tr, the constants ' o,r ', ' ar2 \ ' f l j ' ... for entities of type r\ constants of individual if r = i, constants of proposition if r is of the form nn, and constants of function if r is of the form (n, rh ..., rk)2.

Among the primitive symbols are also to be found the logical symbols ' - i \ '=>', and *V\ corresponding (up to the typological character of the system) to ordinary négation, (material) implication, and universal quantification respectively; the symbol ' V , corresponding to functional abstraction; and parenthèses for punctuation.

The well-formed symbols (or well-formed expressions) will be defined recursively, and, except for the variables and constants, will be, grosso modo, of one of the following forms: '-•/?', '/? =» q\

'(Vx)p', '(Axy ...)p\ Jxy ...'. At the same time the well-formed symbols are defined, they will be assigned a type, and the notion of the real (resp. apparent) occurrence of a variable in such a symbol will be introduced. For whatever values (of the appropriate type) assigned to the variables for the real occurrences they hâve there, a well-formed symbol of type r will correspond to an entity of type r (individual, proposition, or function, depending on the case), and, for this reason, will be called an 'entity symbol', or a 'complète symbol'.

It will be more specifically called a 'formula' if thèse entities are propositions, and a 'term' if not. A complète symbol without any occurrence of a variable will be said to be 'closed', corresponding to a definite entity. A closed formula will be called a 'sentence'. Now, hère is the

Rigorous Définition of the Language L(TTR) 1- Alphabet

1.1- The variables ' K \ where r e Tr and n > 1

2 The abuse of language in thèse explanations will not hâve gone unnoticed: it will never be the letter V itself which occurs as a superscript of a variable or a constant, but the canonical name of a certain r e Tr.

1.2- Possibly, constants ' an\ where r e Tr and n > 1 1.3-*-»* , *=*' , ' V , 'A/ , r , y

2- Complète symbols, formulas, terms, closed formulas (sentences), and closed terms

2.1- The variables and constants with a superscript r are complète symbols of type r. The occurrence of a variable is real in the complète symbol it constitutes.

2.2- If ci and c2 are complète symbols of types nn, nn* respectively, then

r ^ ) ! and r(C l =» C2)1

are complète symbols of types nn, 7tw« respectively, where n" = max(«, «0- The occurrences of variables which were real (resp.

apparent) in C| or c2 remain so in the new complète symbols.

2.3- If c is a complète symbol of type nn, and v a variable of type r having a real occurrence in c, then

T(Vv)cl

is a complète symbol of type nn; where n* = max(n, 1 + od(r)). The occurrences of variables which were real (resp. apparent) in c remain so in the new complète symbol, except for the occurrences of v which were real, which are now apparent.

2.4- If c is a complète symbol of type nn, and v1( ..., vk variables of types rl s ..., rk respectively having real occurrences in c, then

is a complète symbol of type (n\ rj, ..., rk), where n' = max(n, 1 + maxi<;<£od(r£')). The occurrences of variables which were real (resp. apparent) in c remain so in the new complète symbol, except for the occurrences of Vj,..., vk which were real, which are now apparent.

2.5- Ifc, Cj,..., ck are complète symbols of types (n, rl s..., r^), r j , . . . , rk, then

hcx ...c*!

is a complète symbol of type nn. The occurrences of variables which were real (resp. apparent) in c or in one of the c^ remain so in the new complète symbol.

2.6- A complète symbol is a formula or a term depending on whether its type is or is not of the form nn.

2.7- A complète symbol without any real occurrence of a variable is closed; a closed formula is a sentence.

The system TTR would theoretically be obtainable from language L(TTR) by adding axioms and rules of inference corresponding to its intended interprétation. Indeed, besides the axioms and rules of inference obviously expected, Russell introduced certain problematical axioms — the axiom of reducibility, the axiom of infinity, and the axiom of choice — into his own theory of types with the intent of making it powerful enough for the whole of mathematics to be developed within it, and so for verifying the 'logicist' thesis.

4. The Intented Interprétation of the Langage L(TTR)

Let us give a more précise description of the intended interprétation of language L(TTR) as Russell himself might hâve, a more précise description of the hierarchy of individuals, propositions, and functions constituting this interprétation. The two features of this hierarchy it is appropriate to emphasize are as follows.

First, at the bottom of the hierarchy are to be found the 'individuals'. But they are ail worthy of the name only in the typological sensé. For among them are to be found not only things (individuals in the ontological sensé, for example Paul and Mary), but also concepts (for example the predicate rides, the binary relation loves)3. It is through the only opération of prédication4 and, possibly, 3 1°) It was during the Principles period that Russell spoke of 'things' and

'concepts' ('predicates' or Relations'); during the Principia period, he spoke rather of 'particulars' and 'universals', and the change was not just terminological. But that is unimportant for the matters which concern us hère.

2°) Reading Russell, Gôdel seems to hâve had in mind a hierarchy in which the individuals in the typological sensé coincide with the individuals in the ontological sensé. Genetic considérations could shed some light on the reason why this not so in Russell's own work [cf. Rouilhan 1996].

4 Russell does not impose this terminology, which I use hère for want of better.

that of propositional connection, that the propositions without variables — propositions of order 0 — are constructed (for exemple Paul rides, Paul loves Mary).

Second, beyond the individuals, the hierarchy of propositions and functions is built up step by step through the opérations of prédication, connection, quantification, functional abstraction, and application of a function to arguments, without any additional ontological data. In this sensé, which is not at ail anti-realist, one can say that the hierarchy is 'constructive\ There is nothing mysterious about the reason for this constructivity. On the contrary, what would be mysterious, from a Russellian point of view — and this is understandable — would be a proposition or a function which would not be constructed in this sensé. Think about the intended model of the ordinary simple extensional theory of types, about the hierarchy of individuals, classes of individuals, binary (resp. ternary, etc.) relations between individuals, classes of classes of individuals, etc., etc. This hierarchy too has, in an analogous sensé, an obvious constructive character, and, except for questioning the typological character of the theory, no one would think of a class or relation not constructible in this sensé. Again, think about the intended model of Zermelo's set theory [Zermelo 1908, but also and above ail Mirimanoff 1917 and Zermelo 1930], about what is called (deceptively so, since there is nothing typological about the theory in question) the 'cumulative hierarchy of types'. The members of this hierarchy are the individuals (Urelemente) and the sets constructible out of them through the iterated application (extending into the transfinite) of opérations on individuals or sets which hâve been already constructed (séparation, union, etc.). In Mirimanoff s terminology, thèse sets are the 'ordinary' sets, the others would be 'extraordinary'5. Russellian constructivism is not essentially différent.

One can fairly easily picture the Russellian hierarchy in the following way. One first of ail defines the language L*(TTR), independently of any requirement regarding effectivness. For that, one supplies oneself with a set of constants 'flj' in a one-to-one correspondence with the individuals supposed to be given at the lowest level of the hierarchy — and, for each of them, an 'arity':

0 if it corresponds to a thing, 1 if it corresponds to a predicate. n > 2

The prédication in question is not to be confused with the application of a function to arguments.

5 ïndeed, Mirimanoff did not exclude extraordinary sets, but his work was not formalized. For a formai theory including explicitly such sets, see [Aczel 1987].

if it corresponds to a n-ary relation. Literally, the définition of L*(TTR) differs from that of L(TTR) in only the two following ways.

On the one hand, the constants of L*(TTR) are the ' a) ', and there are no others (see below, footnote 6, rule 2.1-a); on the other hand, a new rule assures that one may construct complète symbols of type 7t⁰ using thèse constants or variables of type i (see below, footnote 6, rule2.1-b)⁶.

6 Hère is the définition in extenso ofthe language. L*(TTR):

1- Alphabet

1.1- The variables ' xrn ', where r e Tr and n > l

1.2- The constants * flf- \ with their respective arities n > 0 1.3- 'V, •=>•, 'V\ 'V, '(\ ')•

2- Complète symbols, formulas, terms, closed formulas (sentences), and closed terms 2.1- a) The variables and constants with a superscript r are complète symbols of type r. The occurrence of a variable is real in the complète symbol it constitutes.

b) If z is a n-ary constant, and Z\, ..., zⁿ variables of type i or constants (of whatever arity), then

is a complète symbol of type 7t0. The occurrences of variables are real in it.

2.2- If Ci and c2 are complète symbols of types nw Kn> respectively, then

•"(-ici)"! andl"^ =^ c^

are complète symbols of types Kⁿ, nⁿ» respectively, where n" = max(n, n'). The occurrences of variables which were real (resp. apparent) in cj or c2 remain so in the new complète symbols.

2.3- If c is a complète symbol of type nn, and v a variable of type r having a real occurrence in c, then

is a complète symbol of type nn>, where n' - max(n, 1 + od(r)). The occurrences of variables which were real (resp. apparent) in c remain so in the new complète symbol, except for the occurrences of v which were real, which are now apparent.

2.4- If c is a complète symbol of type nⁿ, and v⁽,..., v^k variables of types r^x,..., r^k respectively having real occurrences in c, then

is a complète symbol of type <«', rh ..., rk), where n'= max(n, 1 + max^- <^ od(rz-))- The occurrences of variables which were real (resp. apparent) in c remain so in the new complète symbol, except for the occurrences of v^ ..., v^ which were real, which are now apparent.

The Russellian hierarchy must, then, be viewed as being the faithful ontological counterpart of the linguistic hierarchy of closed terms and sentences of L*(TTR), or rather of the hierarchy obtained by considering as identical the alphabetic variants of the same complète closed symbol of this linguistic hierarchy. Limiting the constants of L*(TTR) to type i (rule 2.1-a) corresponds to the constructive character of the Russellian hierarchy; the rule 2.1-b corresponds to the propositions of order 0 obtained from individuals through the opération of prédication alone (nothing corresponds to this opération in the language L(TTR), which does not carry out the analysis so far).

5. The Axiom of Reducibility

Adapting otherwise well-known procédures to the particular features of the language L(TTR), one could define the symbols for disjunction, conjunction, (material) équivalence, existential quantification, identity, description, and one could allow oneself to engage in various différent abuses of language, as concerns punctuation, the abstraction operator (to whose use one might prefer, when no confusion is possible, using the circumflex accent à la Russell), type indices (the élimination of which would give way to 'systematically ambiguous' symbols), the marking out of the scopes of descriptions, etc. One could, in that way, recapture the style of Principia. I will myself risk adopting such a style.

I will examine the définition of identity hère, an opportunity for the axiom of reducibility to make its first appearance.

One might think of defining identity by stipulating Leibniz's law for each type:

(1) x = y = mOifl<jx<*fy).

The problem is that quantification cannot range over ail the functions of one argument for which x is a possible argument, but only over

2.5- If c, c1 ( ..., ck are complète symbols of types (n, rx, ..., r^), q , ..., rk , then r^c c j . . . <?*!

is a complète symbol of type nn. The occurrences of variables which were real (resp. apparent) in c or in one of the Cj remain so in the new complète symbol.

2.6- A complète symbol is a formula or a term depending on wether its type is or is not of the form nⁿ.

2.7- A complète symbol without any real occurrence of a variable is closed; a closed formula is a sentence.

those of a spécifie order, for example (this is just an example) those which are of the lowest order compatible with the type of x, and which Russell unfelicitously calls 'predicative'. Whence the Russellian définition [cf. 1908, 245; 1910, 57 and 168-169]:

(2) x = y = D{(Vf)(f\x<*f\y\

where V and 'y' stand for complète symbols of type r e TT, */' for a variable of type (n, r), and where the exclamation mark indicates the predicativity of fi n = 1 + od(r).

One surmises that such a définition will not yield the expected results unless quantifying over the predicative functions of x in some way cornes to the same thing as quantifying over ail the functions of JC, whatever they may be. To make sure of this, one need only let any function of x be coextensive with some predicative function of x. For then, if two entities hâve the same predicative properties, they hâve the same properties plain and simple, they are indiscernible, and can be thought of as identical. The 'axiom of reducibility' will stipulate, in a gênerai way, that any function is coextensive with some predicative function [cf. 1908, 242-243; 1910, 56-58 and 166-16]:

(3) (Y/XBgXV^ ... xk)(fx} ... xk ^ g \xx ... xkX

where 'jef, ..., *xk stand for variables of type rh ..., rk respectively, ' / for a variable of type r = (n, rls..., rk) with n > 1 + m a xl â^ od(r£)), 'g' for a variable of type <m, rL, ..., rk), and where the exclamation mark indicates the predicativity of g: m = 1 + m a x j ^ ^ od(rf)).

Independently of the définition of identity, the axiom of reducibility's support is still needed for the définition of classes of entities, that of binary (resp. ternary, etc.) relations between entities, that of classes of classes of entities, etc., etc. Thèse définitions are 'contextual': they simply consist in indicating how the formulas containing the new symbols — of classes of entities, for example — can be paraphrased into formulas no longer containing them. The classes of entities are not themselves entities; they do not figure in the ontological inventory of the theory of types; they are but 'logical fictions', or, as Russell also says, 'logical constructions' (which one will take care not to confuse with the 'real constructions', so to speak, discussed above). Binary (resp. ternary, etc.) relations between entities, classes of classes of entities, etc., etc. hâve the same

status. I will not bring up Russell's définitions hère7; I will just note that, as in the case of identity, the 'objects' so defined only hâve the expected properties owing to the axiom of reducibility.

Individuals, classes of individuals, binary (resp. ternary, etc.) relations between individuals, classes of classes of individuals, etc., etc. constitute what Russell calls the 'extensional hierarchy'. This hierarchy does not hâve the complexity, the 'ramified' character, as one says, of the intensional hierarchy (that of individuals, propositions and functions). The extensional hierarchy is a 'simple' hierarchy. Its éléments are the very ones one provides oneself with from the outset in the ordinary extensional simple theory of types, hierarchized in identical fashion, but obtained hère in a very roundabout way, and assigned the ontological status of fictional entities.

6. On the Classical Criticisms of the Axiom of Reducibility By appealing to the axiom of reducibility, one can work up and down the extensional hierarchy and everything actually proceeds as in the extensional simple theory of types. And, as has been well- known since at least the end of the twenties, as long as one remains within the extensional simple theory of types, one is safe: ail of classical mathematics can be reconstructed there. Indeed, to do this, one needs an axiom one would hâve liked to do without, the axiom of infinity (not to mention the axiom of choice), but, once one has it, things work out. However, the possibility of a similar reconstruction within the Russellian theory is in turn only opened up through the prior positing of another even more controversial axiom, the axiom of reducibility.

Much has been said about Russell's axiom of reducibility, and sometimes, it was Russell who was the first to speak up.

Poincaré [1909] confessed that he did not really understand what it was about.

Among those who thought they understood, some questioned the nature of the axiom of reducibility. It was no more of a logical axiom than was the axiom of infinity. Wittgenstein [1913b] was, no doubt, the first to attempt to demonstrate it, cryptically, in a letter to Russell at the end of 1913. But, be Wittengstein right or wrong, that did not preclude its being true in our world, and the ability to use it correspondingly.

7 For thèse définitions (revised), cf. [Rouilhan 1996, chap. VI, §§ 19-22].

The axiom of reducibility has also been criticized for being an ad hoc axiom, adopted, not because of any particular self-evidence, but for the sake of the cause. But Russell was well aware of this, and this provided him with an opportunity to reconsider the question of the epistemological grounds for choosing axioms [cf. Russell 1907;

1910, 59-60): logic and mathematics were no less inductive, to his way of thinking, than were the natural sciences, whose principles were not worthy of considération because of any particular self- evidence, but rather because of the expected nature of results that could not be obtained without them.

The axiom of reducibility has also, following Chwistek's lead [1921], been accused of making the Russellian theory of types powerless as regards the semantical paradoxes it was supposed to solve [cf. Chwistek 1922; Copi 1950]. But this accusation turned out to be unfounded [cf. Ramsey 1925; Chwistek 1929; Copi 1970;

Church 1976; Myhill 1979].

Others, following Ramsey's lead [1925], hâve proclaimed the uselessness of an axiom designed to loosen the grip of ramification and, first of ail, of the vicious circle principle, a principle itself useless and even false. This point deserves further considération.

The logicians thus opposed to the vicious circle principle (notably Gôdel [1944, in 1986-??, vol. II, 127-128] and [Quine, 1963, 2d éd., 242-243]) hâve argued that such a principle would only be valid relative to a uni verse of objects "constructed by ourselves".

Their opposition to the principle has gone hand in hand with the idea that mathematical objects "exist independently of our constructions"

just as physical objects do. Conversely, a friend of the vicious circle principle like Feferman [e.g. 1964] has always based his défense of the principle on the idea that abstract objects "do not exist outside of us" and are but "mental constructs". However, in Russell's theory of types, as Quine has noted, the [propositions and] functions are the values of primitive variables and are therefore not "[mental]

constructs", but "entities which are there from the start". Therefore, Russell was wrong to apply the vicious circle principle to them.

At the heart of this criticism is the idea, thereafter accepted, that the vicious circle principle would justifiably be party to constructivism in the anti-realist, to be more spécifie the mentalistic sensé, in which it is a question of it in this matter. It is clear that Russell did not think that at ail, and he was quite right in that. Even though he hardly explained it, the vicious circle principle for propositions and functions had nothing to do, to his way of thinking, with any possible character of mental construction of thèse entities, but rather, no doubt, with the extrême severity of the identity criteria thèse intensions were supposed to meet. For the idea that an intension

satisfying such identity criteria might be the value of its own apparent variables turns out, upon réfection, to be hardly less shocking than the idea that it might be one of its own constituents, or the idea that a class or a set might be one of its éléments, or an élément of one of its éléments, etc.

This latter kind of possibility is known to be ruled out in both the intended model of the extensional simple theory of types and the intended model of Zermelo's set theory; I will call the heuristic principle for this exclusion the 'principle of foundation', whose formai counterpart for set theory goes by the name 'axiom of foundation'. The concept of set in the sensé intended by Zermelo's set theory is known by the name 'itérative concept of set'. I will use the term 'itérative concept of extension' to refer to both the concept of class or relation in the sensé intended by the extensional simple theory of types and the concept of set in the sensé intended by Zermelo's theory. Finally, since Poincaré [1905-06], any notion in keeping with the vicious circle principle is called 'predicative'. I will therefore say: for sufficiently strict identity criteria, a predicative concept of intension is just as natural as an itérative concept of extension, and the vicious circle principle as the principle of foundation.

7. GôdePs Criticism : on the Incompatibility of the Axiom of Reducibility with the Axiom of Infinity in any Constructive Interprétation of the Language of the Russellian Theory of Types

It remains for me to bring up a last criticism, which, if it is justified, is truly invalidating. This is Gôdel's criticism.

There is something extremely disturbing about the idea that the axiom of reducibility and the axiom of infinity make it possible to reconstruct classical mathematics within the ramified theory of types, i.e. to reconstruct, within a logic supposed to be predicative (that is to say in keeping with the vicious circle principle), mathematics which, as we well know nowadays, is not! One suspects that there is something fundamentally wrong with the axiom of reducibility, that, together with the axiom of infinity, the axiom of reducibility reintroduces behind the scènes what ramification, and in the first place the vicious circle principle, hâve ruled out in principle, namely impredicativity. But how can this be with an axiom which obviously does not involve any type mistake?

But it is not in the direction of the language that one has to look. It is to its interprétation. As an intended interprétation, it must be constructive (in the realist sensé); furthermore, in the case under considération, it must satisfy the axiom of reducibility and the axiom

of infinity. But can an interprétation satisfy both thèse axioms simultaneously while retaining this constructive character? That is the question. And Gôdel's answer [1944] is no. (Naturally, it was not exactly to the question I am now asking that Gôdel answered no. He did not formalize his idea, and he seems to hâve had in mind a theory slightly différent from the one I hâve drawn from reading Russell)8.

Hère is what Gôdel says:

[The axiom of reducibility] in essence already mean[s] the existence in the data of the kind of objects to be constructed [...]. In the first édition of Principia, [...] the constructivisme attitude was, for the most part, abandoned, since the axiom of reducibility [...] together with the axiom of infinity makes it absolutely necessary that there exist primitive predicates of arbitrarily high types [...]. [T]he axiom of reducibility [...]

(in the case of infinitely many individuals) is demonstrably false unless one assumes [...] the existence of classes or of infinitely many 'qualitates occultae¹. [Gôdel 1944, 142, 143 and 152]

In the présence of the axiom of infinity, if Gôdel is right — and I believe he is right — the axiom of reducibility is scarcely worth anything more, ail things considered, than a contradiction. I do not know what proof Gôdel had in mind. Perhaps the publication of the Nachlass will provide the answer. The Collected Works currently being published do not, for the time being, promise anything of the like.

Even worse, Charles Parsons, in his introduction to Gôdel's 1944 article, thinks one can dismiss any such proof beforehand:

Goder s remarks about the axiom of reducibility show lack of sensitivity to the essentially intensional character of Russell's logic;

the fact that every propositional function is coextensive with one of lowest order does not imply "the existence in the data of the kind of objects to be constructed", if the objects in question are concepts or propositional functions rather than classes . [...] Russell himself was closer to the mark in saying that the axiom accomplishes "what common sensé effects by the admission of classes." [...] This insensitivity is quite common in commentators on Russell, but is somewhat surprising in Gôdel [...]. [Parsons 1990, 112-113]

Be Gôdel right or wrong, the proceedings Parson institutes against him for having missed the essentially intensional character of Russell's logic is absurd. Parsons seems to believe that Gôdel thinks of the axiom of reducibillity as essentially affirming, for any function of one argument, for example, the existence of the corresponding

8 Cf. above, footnote 3, 2°.

class. He seems to believe that the 'objects to be constructed* Gôdel speaks of can be but classes. He seems to forget Russell's realist constructivism, so to speak, regarding propositions and functions, and retain only Russell's logical constructivism (or fictionalism) regarding classes and relations, to which Gôdel does not limit himself at ail. Lack of compréhension is quite common among commentators on Gôdel, but it is somewhat surprising in Parsons, even though it is true that Gôdel himself failed explicitly to make the radical distinction between the two kinds of constructivism called for — unless it is primarily a matter of not understanding Russell himself9.

I will say, to conclude, why I believe Gôdel was right: by analogy with an otherwise better known situation. Let us forget about Russell. Let us take as our individuals the set of whole numbers N.

Let us define M0 as the class of arithmetical sets; and M! as the class of sets definable by a condition (without parameters) in which quantification is relative to N or to M0. The question of knowing whether, in a constructive interprétation of L(TTR) satisfying the axiom of infinity, every function of order 2 of an individual argument, for example, is coextensive with a predicative function (in the sensé involved in the axiom of reducibility) is analogous to the question wether M0 = Ml. The answer is no. By transfinite induction, one can, indeed, define an ever growing séries of classes Ma leading up, at a certain a>j"th step, to the class A1, of so-called

'hyperarithmetical' sets: Mœ l = Ap On the other hand, we know that there exist hyperarithmetical sets which are not arithmetical:

M0 ?fc A\ (thèse results are due to Kleene [1955a; 1955b; 1959]). It follows from this, given the définition of Ma, that, already, M0 ^ M j . It now remains to verify that this démonstration that M0 * M!

can actually be transposed into a démonstration for L(TTR) of Gôdel's thesis. In this form, the thesis at least has the merit of being mathematically précise, and one has an idea of how to demonstrate it. But that does not mean, unfortunately, that what cornes next is a routine affair.

If the thesis I am defending in the wake of Gôdel is correct, it deals a death blow to the Russellian version of the ramified theory of types with the axiom of reducibility. Those wishing to rescue the

9 The above formulation of my criticism of Parsons corrects the one I gave in [1996, 274].

theory against Russell himself would hâve but to forgo the constructive character (in the realist sensé) the latter required of the hierarchy of propositions and functions, forgo the intelligibility they had in the beginning, and admit 'qualitates occultae\ to use Gôdel's expression, and even, more specifically, "qualitates occultae" of arbitrarily high types. Or better, if possible, contest the 'occult' character of thèse qualities¹⁰.

Références

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